JP2006-336816A and JP2007-078004A, published by the Japan Patent Office in 2006 and 2007, respectively, disclose a damping device comprising an upper chamber and a lower chamber defined as working chambers by a piston housed in a cylinder, a first passage that penetrates the piston in order to connect the upper chamber and the lower chamber under a predetermined flow resistance, a pressure chamber formed in the piston, a free piston that partitions the pressure chamber into an upper pressure chamber that communicates with the upper chamber and a lower pressure chamber that communicates with the lower chamber, and a coil spring that supports the free piston elastically.
The pressure chamber does not connect the upper chamber and the lower chamber directly, but when the free piston displaces, a volume ratio between the upper pressure chamber and the lower pressure chamber varies. More specifically, a working fluid moves between the upper chamber and the upper pressure chamber and between the lower chamber and the lower pressure chamber. As a result, the pressure chamber functions as a second passage that substantially connects the upper chamber and the lower chamber.
When a differential pressure between the upper chamber and the lower chamber, generated as the damping device expands and contracts, is set as P and a flow rate flowing out of the upper chamber is set as Q, a transfer function G (s) of the differential pressure P relative to the flow rate Q is determined from a following Equation (1).
                              G          ⁡                      (            s            )                          =                                            C              ⁢                                                          ⁢                              1                ·                1                                      +                                          A                2                            ·                                                                    (                                                                  C                        ⁢                                                                                                  ⁢                        2                                            +                                              C                        ⁢                                                                                                  ⁢                        3                                                              )                                    ⁢                  s                                K                                                          1            +                                          A                2                            ⁢                                                                    (                                                                  C                        ⁢                                                                                                  ⁢                        1                                            +                                              C                        ⁢                                                                                                  ⁢                        2                                            +                                              C                        ⁢                                                                                                  ⁢                        3                                                              )                                    ⁢                  s                                K                                                                        (        1        )                            where:        Q1 is a flow rate of he firs passage;        C1 is a coefficient indicating a relationship between he differential pressure P and he flow rate Q1;        P1 is a pressure of he upper pressure chamber;        P2 is a pressure of he lower pressure chamber;        Q2 is an inflow flow rate from he upper chamber into he upper pressure chamber and an outflow flow rate from he lower pressure chamber into he lower chamber;        C2 is a coefficient indicating a relationship between he flow rate Q2 and a difference between he differential pressure P and he pressure P1;        C3 is a coefficient indicating a relationship between he pressure P2 and he flow rate Q2;        A is a pressure receiving surface area of he free piston;        K is a spring constant of he coil spring; and        s is a Laplacian operator.        
By substituting jω for the Laplacian operator s in Equation (1) and determining an absolute value of a frequency transfer function G (jω), Equation (2) is obtained.
                                                    G            ⁡                          (                              j                ⁢                                                                  ⁢                ω                            )                                                =                                                                                                  C                    ⁢                                                                                  ⁢                    1                    ⁢                                                                                  ⁢                                          K                      4                                                        +                                                                                    K                        2                                            ·                                              A                        4                                                              ⁢                                                                  {                                                                              2                            ·                                                          (                                                                                                C                                  ⁢                                                                                                                                          ⁢                                  2                                                                +                                                                  C                                  ⁢                                                                                                                                          ⁢                                  3                                                                                            )                                                        ·                                                          (                                                                                                C                                  ⁢                                                                                                                                          ⁢                                  1                                                                +                                                                  C                                  ⁢                                                                                                                                          ⁢                                  2                                                                +                                                                  C                                  ⁢                                                                                                                                          ⁢                                  3                                                                                            )                                                                                +                                                      C                            ⁢                                                                                                                  ⁢                                                          1                              2                                                                                                      }                                            ·                                                                                                                                                                ω                    2                                    +                                                            A                      5                                        ·                                                                  (                                                                              C                            ⁢                                                                                                                  ⁢                            2                                                    +                                                      C                            ⁢                                                                                                                  ⁢                            3                                                                          )                                            2                                        ·                                                                  (                                                                              C                            ⁢                                                                                                                  ⁢                            1                                                    +                                                      C                            ⁢                                                                                                                  ⁢                            2                                                    +                                                      C                            ⁢                                                                                                                  ⁢                            3                                                                          )                                            3                                        ·                                          ω                                              4                        ⁢                                                                                                  ⁢                                                  1                          2                                                                                                                                                                            K              2                        +                                          A                4                            ·                                                (                                                            C                      ⁢                                                                                          ⁢                      1                                        +                                          C                      ⁢                                                                                          ⁢                      2                                        +                                          C                      ⁢                                                                                          ⁢                      3                                                        )                                3                            ·                              ω                2                                                                        (        2        )            
As is evident from Equations (1) and (2), a frequency characteristic of the transfer function of the differential pressure P relative to the flow rate Q has a cutoff frequency Fa expressed by a following Equation (3) and a cutoff frequency Fb expressed by a following Equation (4).
                    Fa        =                  K                      {                          2              ≠                                                A                  2                                ·                                  (                                                            C                      ⁢                                                                                          ⁢                      1                                        +                                          C                      ⁢                                                                                          ⁢                      2                                        +                                          C                      ⁢                                                                                          ⁢                      3                                                        )                                                      }                                              (        3        )                                Fb        =                  K                      {                          2              ≠                                                A                  2                                ·                                  (                                                            C                      ⁢                                                                                          ⁢                      2                                        +                                          C                      ⁢                                                                                          ⁢                      3                                                        )                                                      }                                              (        4        )            
Referring to FIG. 21, a transfer gain is substantially equal to C1 in a frequency region F<Fa, gradually decreases from C1 to C1·(C2+C3)/(C1+C2+C3) in a region Fa≦F≦Fb, and becomes constant in a region F>Fb. In other words, the frequency characteristic of the transfer function of the differential pressure P relative to the flow rate Q has a large transfer gain in a low frequency region and a small transfer gain in a high frequency region.
Referring to FIG. 22, this damping device generates a large damping force relative to low frequency vibration input and a small damping force relative to high frequency vibration input. This characteristic contributes to realization of a favorable level of passenger comfort when the damping device is used in a vehicle. The reason for this is that low frequency vibration input such as centrifugal force that acts as the vehicle turns is absorbed by the large damping force, while high frequency vibration input such as that generated by irregularities on a road surface as the vehicle travels is damped by the small damping force.